One of the main thrusts of research in the topology of dynamical systems is an investigation into the topological structure of sets left invariant under a dynamical system, such as attractors. We propose investigating the topological structure of a family of models for the non-hyperbolic H'{e}non attractors: inverse limit spaces generated by unimodal maps of the interval. A celebrated conjecture in the topological theory of inverse limits is the Ingram Conjecture which states that the kneading sequence is a topological invariant for inverse limits generated by unimodal maps. Our aim is to classify a large family of inverse limits generated by unimodal maps, namely the inverse limit spaces generated by tent maps with a sparse postcritical orbit. This will be a major step towards a proof of the Ingram Conjecture.
The study of chaotic dynamical systems has surged in importance in the last few decades. One of the main avenues of research is into the delicate structure of chaotic attractors. A major stumbling block in this endeavor is that usually we do not have a precise mathematical description for these attractors, but rather we have only numerical evidence. However, there is a family of models for certain chaotic attractors that do have a precise mathematical description: inverse limit spaces of unimodal maps. Understanding the structure of these inverse limit spaces more fully will lead to a better understanding of chaotic attractors. This proposal represents an international collaboration with the University of Zagreb, Croatia. The aim of the collaboration is to fully describe the structure of this family of models for certain chaotic attractors.
